Mathematics Programme Area

WilberforceCollege

Mathematics Programme Area

Resources for Unit 1

Skills 1 +

Linear Functions

Susan Wall

WilberforceCollege

Saltshouse Road

Hull

HU8 9HD

01482 711688

Skills 1

These activities are not necessarily in the order in which they should be done. They are intended to give ideas and to be adapted to suit the needs of different learners rather than be used exactly as written. Several activities may cover the same objective – this is to give choice rather than repetition. Activities are not intended to cover every aspect of the scheme of work. They are opportunities for a more interactive way of learning where appropriate.

Most activities are more effective if done in pairs or groups.

At the end of each activity, particularly matching ones, students should be encouraged to write up one or two of the examples with detailed reasons for future reference.

On complex diagrams all the individual parts have been grouped together using the ‘draw’ toolbar. This enables the whole page to be moved around. However if one part needs changing then highlight the page, click on draw and then ungroup.

Activity 1

The templates for these are in separate files – Algebra1, Algebra2, Algebra3 as they are done on Publisher.

This is an algebraic jigsaw for multiplying out brackets. There are 3 sheets of equilateral triangles that fit together to make a hexagon. It can be done in groups of 3 or 4 as it is quite large. It is an excellent activity for the first lesson as it brings to the fore many misconceptions about brackets that can be discussed – go round and ask why certain pieces have been fitted together. It also gets students used to working together in a non-threatening way – they are collaborating but there does not have to be a huge amount of conversation as each member of the group can work on individual pairs to start with and gradual cooperation emerges.

Activity 2

Use whiteboards. Tell each student to write down say 3x + 1. Then tell them to add 4, then add 6x, then double it , then square it …. Compare results and discuss differences.

The starter could be more complex such as 2(3x + 4) and then add 2x etc.

Repeat with an equation such as x = 4 and give instructions to be performed on both sides.

See what happens – it is amazing what they will come up with! But it is good practice for algebraic manipulation with everyone joining in.

Activity 3

Ask students to work through “Substitution in Formulas”. Discuss the errors made pointing out the importance of order of operations and putting negative numbers in brackets when raising to a power.

Activity 4

“Using Function Notation” is a multichoice set of questions. It is aimed at highlighting and bringing to the fore common errors and misconceptions eg 8 – (-1)² as 8 - -1²and so getting 9 instead of 7 but both answers are there.

This could be done as a whole class activity in groups with each group coming up with the answer a, b, c on their whiteboards and points given for correct answers and justifications.

Activity 5

Open ended tasks can be given using function notation. Give the students an equation such as f(3) = 7 and ask them to find as many functions as they can for f(x).

Then negative numbers and fractions can be included to make them think more.

This could then lead onto finding possible f(x) and g(x) such that g(5) = f(3) etc.

Activity 6

Using sheet “Linear Equations”, ask students to mark it, correct it and write some comment to the student whose work it is to explain what they have done wrong in each case.

Activity 7

Use the sheet “Matching Up” to practise more substitution via a new context using sigma notation. These could be cut onto card for a hands on activity. At the end they should write up a couple with detailed explanation about how to evaluate an expression involving sigma notation.

Activity 8

As an extension exercise linking all the skills covered together functions f(x) and g(x) could be defined and students asked to solve equations such as f(x + 1) = g(x) or f(3x) = g(x + 1) or f(x – 1) = 2g(5) etc. These use function notation, solving linear equations, multiplying out brackets and substituting in formulas.

Activity 3 Substitution in Formulas

These answers are wrong. Why?

(a)

(b)

(c)

(d)

(e)

(f)

Activity 4Using Function Notation

Activity 6Linear Equations

Mark these correct, mark those that are wrong and write helpful comments in the box

1 / / 2 /
Comments: / Comments:
3 / / 4 /
Comments: / Comments:
5 / / 6 /
Comments: / Comments:

Activity 7Matching Up

Linear Functions

Most activities are more effective if done in pairs or groups.

At the end of each activity, particularly matching ones, students should be encouraged to write up one or two of the examples with detailed reasons for future reference.

Activity 1

This is a whole class introduction (and revision from GCSE) to work done on straight lines. Write y = 2x + 1 in the middle of the board. Ask for anyone to contribute anything that they know about that function. Hope to get that it is a straight line, something about y intercept, something about gradient and something about points that it goes through. A little prompting is in order here! Just write everything on the board without explanation.

Then discuss and ask for explanation for each property. Link gradient as coefficient of x with a gradient triangle from two of the points given. Link y intercept with the idea of substituting x = 0 and then solving for y. This leads onto the x intercept as solving for x when y = 0.

Give everyone (or in pairs) a piece of A3 paper and an equation of the form y = mx + c. Everyone can have a different equation and so can check each others. Then ask them to produce a diagram like the one below for their equation.

Then repeat by givng everyone an equation of the form 3x + 4y = 12. This should produce some interesting discussions particularly for those who say it has gradient 3 and then find a different answer from their gradient triangle.

Activity 2

Quick activity on rearranging equations using “Match the Equations”. Students should match them up in threes along with a graph. The graphs have some co-ordinates marked on and students could can identify equations by checking the co-ordinates or by using y = mx + c. They can work in pairs for this. Rather than just giving the answers ask students to explain why equations are equivalent and justify their choice of graph.

Activity 3

More matching equations to graphs. Again students can work in pairs.

Students should match the equations to the graphs write down the gradient and intercepts on both axes for each line. When discussing answers draw out that parallel lines can be recognised as having the same gradient.

Activity 4

Draw perpendicular lines on square paper and investigate perpendicular gradients. Practise using mini whiteboards.

Activity 5

Put some equations on the board or give out a set of cards (Sheet: Sort into Categories) to each pair of students. Ask questions – if using cards then they can hold up the card with the graph on, otherwise write the equation on a mini – whiteboard. Questions could include:

Which lines have gradient 3?

Which lines are perpendicular?

Which go through (2, 1)

Which are parallel? etc

Activity 6

Using the same cards as activity 5, students should be asked to sort them according to their own categories. The choices can then be discussed or categories could be given eg same y intercept, parallel lines, perpendicular lines, find 2 with the same x intercept, find 2 with the same y intercept. For each category students should then be given a blank piece of card and asked to add another equation that fits and asked to justify their choice.

Activity 7

Mini whiteboards can be used to consolidate by sketching graphs one at a time on the board (without any numbers marked on) e.g.

and asking for possible equations for each one and encouraging equations to be as different as possible. Students must not have the same answer as the person sitting next to them!

Then practice sketching straight lines. Give an equation, students sketch it roughly and show it - use whiteboards. Give another and repeat. Just look for slope in correct direction and correct intercepts marked. Include ones such as 2x + 3y= 7.

More challenging questions can be used such as:

Activity 8

Other questions to use with mini whiteboards are:

Give me an example of a line parallel to the line y = 2x – 4.

Give me an example of a line that is perpendicular to the line y =7 - 3x.

Give me an example of 2 parallel lines.

Give me an example of 2 perpendicular lines.

Give me an example of a line with a gradient of 4.

Give me 2 lines that intersect at (3, 1).

Give me an example of a line that passes through the point (2, 3).

Activity 9

Every student chooses a gradient and a point and writes it on a piece of card (or paper). Then the students find the equation of their line. They then write down the coordinates of 2 points on their line on another piece of paper and pass it to another student who has to find its equation. Answers can be checked by returning them to the original student. This idea can also be used to find lines that are perpendicular to a given line. Also ones that go through a given point.

Another variation is students filling in a point and gradient of their own choice on a copy of the sheet below. They then fill in the rest and the sheets are passed around the class to other students for checking:

My point is: ………………………

My gradient is: ……………………….

My equation is: …………………………………………………………………..

A parallel line is: …………………………………………………………………

A perpendicular line is: ……………………………………………………………..

Activity 10

Photocopy the second sheet of “Pairs” with properties on, onto A3 paper. Then students have to find 2 lines from the first sheet that fit into each category. They stick them on and fill in a property for the two that do not fit anywhere. They could add one of their own to each set when they have finished.

Activity 11

Students should read the solution on the sheet "Perpendicular Bisectors” and describe what is happening on each line of the solution and write it next to the line.

Activity 12

Using the sheet “Jigsaw Problem” students have to cut out the stages of the solution and then sort the solution of a problem into the correct order. They can then stick them down and add some comments.

Activity 13

Students mark and correct the sheet “Intersection of Lines”.

Students could then make up their own pair of lines (and solve them) then give them to their partner to solve. Then they can compare answers!

Activity 14

This is a straightforward matching activity with equations and properties of straight lines. The equations are 2y = x + 6, y = 3 – 2x, y = 3x + 2,

y = 2x – 3, 3y + 2x = 12 and y = 4x and the properties are on sheet “Properties of Straight Lines”. These equations can be written onto an A3 sheet and the properties can be cut out and stuck with the appropriate equation along with the explanation.

Harder or easier versions of the properties can be used as appropriate.

Activity 15

"Solving Inequalities" starts with matching some inequalities with regions on graphs and then sees that problems can be solved using a range of different strategies:

Solve the equality and use the graphs to get the correct inequality.

Solve the equality and use a test number to get the correct inequality.

Rearrange the inequality with care.

Activity 16

Suggestions for writing:

Describe how you:

Decide if 2 lines are parallel.

Decide if 2 lines are perpendicular.

Find a midpoint between 2 points.

Find the y intercept of a line.

Find the x intercept of a line.

Find where 2 lines intersect.

Write a series of bullet points to show how to find the equation of the

straight line that joins two given points.

Activity 2Matching the Equations

Match the equations to the line (3 for each) and be prepared to have to give a reason for your choice!

Activity 3

Label the lines with these equations:

Activity 5 Sort into Categories

y = 4x + 1

/ y – 4x = 3

y + 3x = 1

/ 2y – 8x = 7

y + 4x = 1

/ y = 3x + 1
y – 3x = 3 / y = 7 – 3x
2y = 7 – 8x / y + 3x = 3
4y + x = 1 / 3y + x = 3
4y = 7 + x / 3y = 1 + x

Activity 10Pairs

y = 4x + 4 4y = x + 3
y = 8x – 3y + 4x + 6 = 0
3y =2x – 8 y + 6x = 11
y + 8x = 6 2y + 8 = 3x
2y + x = 4 2y = 8x + 3
y = 6x – 4 y + x + 8 = 0

These lines are parallel.These lines are perpendicular.

These lines have the same y intercept.These lines have the same x intercept.

These lines both go through the point (1, 5).These lines …..

Activity 11Perpendicular Bisectors

Question:Find the perpendicular bisector of the line joining the points (-2, 11) and (4, -7).

Solution:Explanation:

Activity 12Jigsaw Problem

Question:Three points have co-ordinates A(1, 7), B(7, 5) and C(0, -2). Find: (a) the equation of the perpendicular bisector of AB.

(b)the equation of the line BC

(c)the point of intersection of these two lines.

------

Y = x - 2

------

4 = 2x x = 2

------

y = 3x + c

------

y = 3x - 6

------

Point is (2, 0)

------

Gradient = -2/6 = -1/3

------

6 = 3 x 4 + c  c = -6

------

x - 2 = 3x - 6

------

-2 = c

------

Point is (4, 6)

------

Gradient is 3

------

Y = x + c

------

Gradient is 1

Activity 13Intersection of Lines

Find where the following pairs of lines intersect:

Solutions – please mark and correct:

Activity 14Properties of Straight Lines

y intercept is 2 / Parallel to y = 2x + 7
Passes through (0, 0) / Passes through (1, 5)
Passes through (0, 4) / y intercept is –3
x intercept is -2
3 / y intercept is 4
Gradient is –2 / Gradient is 2
Parallel to y = 3x – 1 / Perpendicular to y = 3 – 2x
Perpendicular to y = 5 – x
4 / Passes through (3, 2)
Gradient is 3 / Perpendicular to y = 3x + 1
2
x intercept is –6 / x intercept is 6

Activity 15Solving Inequalities

2x + 1 > 5x - 14x + 6 > 4x + 9

Which is which?

x + 3 > -4x – 127 – x < x + 3

x > 2x < 5Which do these belong to?x > -3x < -1

How would you solve How does your solution relate to the graphs below?

Test some numbers to check your solution.

Explain how this solution set was reached.

Illustrate the solution set using the graphs of and .

Does

or does ?

Which way round is correct and why? Test the inequality with some numbers to check your answer.